Theorem fmf 23865

Description: Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fmf	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))

Proof of Theorem fmf

Dummy variables 𝑓 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7402	. . . 4 ⊢ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))) ∈ V
2		eqid 2729	. . . 4 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
3	1, 2	fnmpti 6643	. . 3 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) Fn (fBas‘𝑌)
4		df-fm 23858	. . . . . 6 ⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)))))
5	4	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))))))
6		dmeq 5857	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
8		fdm 6679	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
9	8	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → dom 𝐹 = 𝑌)
10	7, 9	sylan9eqr 2786	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → dom 𝑓 = 𝑌)
11	10	fveq2d 6844	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → (fBas‘dom 𝑓) = (fBas‘𝑌))
12		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
13		imaeq1 6015	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓 “ 𝑦) = (𝐹 “ 𝑦))
14	13	mpteq2dv 5196	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
15	14	rneqd 5891	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
16	12, 15	oveqan12d 7388	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
17	16	adantl 481	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
18	11, 17	mpteq12dv 5189	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
19		elex 3465	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
20	19	3ad2ant1 1133	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ V)
21		fex2 7892	. . . . . 6 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → 𝐹 ∈ V)
22	21	3com13 1124	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ V)
23		fvex 6853	. . . . . . 7 ⊢ (fBas‘𝑌) ∈ V
24	23	mptex 7179	. . . . . 6 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) ∈ V
25	24	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) ∈ V)
26	5, 18, 20, 22, 25	ovmpod 7521	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
27	26	fneq1d 6593	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ↔ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) Fn (fBas‘𝑌)))
28	3, 27	mpbiri 258	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
29		simpl1 1192	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋 ∈ 𝐴)
30		simpr 484	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
31		simpl3 1194	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌⟶𝑋)
32		fmfil 23864	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
33	29, 30, 31, 32	syl3anc 1373	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
34	33	ralrimiva 3125	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋))
35		ffnfv 7073	. 2 ⊢ ((𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋) ↔ ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ∧ ∀𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) ∈ (Fil‘𝑋)))
36	28, 34, 35	sylanbrc 583	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ↦ cmpt 5183  dom cdm 5631  ran crn 5632   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  fBascfbas 21284  filGencfg 21285  Filcfil 23765   FilMap cfm 23853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-fbas 21293  df-fg 21294  df-fil 23766  df-fm 23858

This theorem is referenced by:  rnelfm  23873